Taylor coefficients and zeroes of entire functions of exponential type

Abstract

Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} ω_n \frac{z^n}{n!} \] with unimodular coefficients $|{\omega }_n|=1$. We show that either the counting function $n_F(r)$ of zeroes of $F$ grows linearly at infinity, or $F$ is an exponential function. The same conclusion holds if only a positive asymptotic proportion of the coefficients ${\omega }_n$ is unimodular. This significantly extends a classical result of Carlson (1915). The second result requires less from the coefficient sequence $\omega$, but more from the counting function of zeroes $n_F$. Assuming that $0<c\le |{\omega }_n|\le C<\infty$, $n\in {\mathbb{Z}}_{+}$, we show that $n_F(r)=o(\sqrt{r})$ as $r\to \infty$, implies that $F$ is an exponential function. The same conclusion holds if, for some $\alpha <1/2$, $n_F(r_j)=O(r_j^{\alpha })$ only along a sequence $r_j\to \infty$. Furthermore, this conclusion ceases to hold if $n_F(r)=O(\sqrt{r})$ as $r\to \infty$.

35 pages

References in corpus (3)

• Stable polynomials and crystalline measures
• Perturbed Fourier uniqueness and interpolation results in higher dimensions
• Crystalline temperate distributions with uniformly discrete support and spectrum